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Spectral Singularities, Biorthonormal Systems, and Complex Point Interactions

Alí Mostafazadeh, Hossein Mehri‐Dehnavi

Published Jan 23, 2009• 0 citations
OpenAlex

Abstract

A curious feature of complex scattering potentials v(x) is the appearance of spectral singularities. We offer a quantitative description of spectral singularities that identifies them with an obstruction to the existence of a complete biorthonormal system consisting of the eigenfunctions of the Hamiltonian operator, i.e., − d2 dx 2 +v(x), and its adjoint. We establish the equivalence of this description with the mathematicians ’ definition of spectral singularities for the potential v(x) = z−δ(x + a) + z+δ(x − a), where z ± and a are respectively complex and real parameters and δ(x) is the Dirac delta-function. We offer a through analysis of the spectral properties of this potential and determine the regions in the space of the coupling constants z ± where it admits bound states and spectral singularities. In particular, we find an explicit bound on the size of certain regions in which the Hamiltonian is quasi-Hermitian and examine the consequences of imposing PT-symmetry. PACS numbers: 03.65.-w

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A curious feature of complex scattering potentials v(x) is the appearance of spectral singularities. We offer a quantitative description of spectral singularities that identifies them with an obstruction to the existence of a complete biorthonormal system consisting of the eigenfunctions of the Hamiltonian operator, i.e., − d2 dx 2 +v(x), and its adjoint. We establish the equivalence of this description with the mathematicians ’ definition of spectral singularities for the potential v(x) = z−δ(x + a) + z+δ(x − a), where z ± and a are respectively complex and real parameters and δ(x) is the Dirac delta-function. We offer a through analysis of the spectral properties of this potential and determine the regions in the space of the coupling constants z ± where it admits bound states and spectral singularities. In particular, we find an explicit bound on the size of certain regions in which the Hamiltonian is quasi-Hermitian and examine the consequences of imposing PT-symmetry. PACS numbers: 03.65.-w
A curious feature of complex scattering potentials v ( x ) is the appearance of spectral singularities. We offer a quantitative description of spectral singularities that identifies them with an obstruction to the existence of a complete biorthonormal system consisting of the eigenfunctions of the Hamiltonian operator, i.e., , and its adjoint. We establish the equivalence of this description with the mathematicians' definition of spectral singularities for the potential v ( x ) = z − δ( x + a ) + z + δ( x − a ), where z ± and a are respectively complex and real parameters and δ( x ) is the Dirac delta function. We offer a through analysis of the spectral properties of this potential and determine the regions in the space of the coupling constants z ± where it admits bound states and spectral singularities. In particular, we find an explicit bound on the size of certain regions in which the Hamiltonian is quasi-Hermitian and examine the consequences of imposing -symmetry.
We explore the Hamiltonian operator H=-d^2/dx^2 + z \\delta(x) where x is real, \\delta(x) is the Dirac delta function, and z is an arbitrary complex coupling constant. For a purely imaginary z, H has a (real) spectral singularity at E=-z^2/4. For \\Re(z)<0, H has an eigenvalue at E=-z^2/4. For the case that \\Re(z)>0, H has a real, positive, continuous spectrum that is free from spectral singularities. For this latter case, we construct an associated biorthonormal system and use it to perform a perturbative calculation of a positive-definite inner product that renders H self-adjoint. This allows us to address the intriguing question of the nonlocal aspects of the equivalent Hermitian Hamiltonian for the system. In particular, we compute the energy expectation values for various Gaussian wave packets to show that the non-Hermiticity effect diminishes rapidly outside an effective interaction region.
We explore the Hamiltonian operator , where is the Dirac delta function and z is an arbitrary complex coupling constant. For a purely imaginary z , H has a spectral singularity at . For Re( z ) < 0, H has an eigenvalue at E = − z 2 /4. For the case that Re( z ) > 0, H has a real, positive, continuous spectrum that is free from spectral singularities. For this latter case, we construct an associated biorthonormal system and use it to perform a perturbative calculation of a positive-definite inner product that renders H self-adjoint. This allows us to address the intriguing question of the nonlocal aspects of the equivalent Hermitian Hamiltonian for the system. In particular, we compute the energy expectation values for various Gaussian wave packets to show that the non-Hermiticity effect diminishes rapidly outside an effective interaction region.
Spectral singularities are spectral points that spoil the completeness of the eigenfunctions of certain non-Hermitian Hamiltonian operators. We identify spectral singularities of complex scattering potentials with the real energies at which the reflection and transmission coefficients tend to infinity, i.e., they correspond to resonances having a zero width. We show that a waveguide modeled using such a potential operates like a resonator at the frequencies of spectral singularities. As a concrete example, we explore the spectral singularities of an imaginary PT-symmetric barrier potential and demonstrate the above resonance phenomenon for a certain electromagnetic waveguide.
Spectral singularities have been extensively studied over the last one and half decades for different non-Hermitian potentials in non-Hermitian quantum mechanics. The nature of spectral singularities has not been studied for the case of quaternionic potential. In the present work, we perform an analytical study on scattering from a quaternionic point interaction represented by a delta function. New features of spectral singularities are observed, which are different than the case of a complex (non-quaternionic) point interaction. The most notable difference is the occurrence of spectral singularity from the lossy point interaction, which is forbidden in the case of standard non-Hermitian quantum mechanics.